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Concept

Permutation matrix

Summary

In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say P, represents a permutation of m elements and, when used to multiply another matrix, say A, results in permuting the rows (when pre-multiplying, to form PA) or columns (when post-multiplying, to form AP) of the matrix A. Definition Given a permutation π of m elements, :\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace represented in two-line form by :\begin{pmatrix} 1 & 2 & \cdots & m \ \pi(1) & \pi(2) & \cdots & \pi(m) \end{pmatrix}, there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the m × m identity matrix, Im, either permute the columns or permute the rows, according to π. Both methods of defin

Official source

https://en.wikipedia.org/wiki/Permutation_matrix

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We consider the problem of providing delay bounds to reserved traffic in high-speed input-queued switches. We assume that the matrix of bandwidth demands is known and we use the now standard approach of decomposing this matrix into a convex combination of permutation matrices. Our problem therefore reduces to the problem of constructing a schedule for these permutation matrices. In this paper we derive delay bounds for four algorithms that are based on probabilistic techniques. For each algorithm we first place tokens randomly in continuous time for each permutation matrix. If the

n

th token that appears corresponds to permutation matrix

M_k

then we schedule matrix

M_k

in the

n

th time slot. The algorithms differ in how the random token processes are defined. For two of the algorithms we are able to perform a derandomization so as to obtain deterministic schedules. We show through numerical computation that in many situations the resulting delay bounds are smaller than the previously best-known delay bounds of Chang, Chen, and Huang.

2003

Scheduling Reserved Traffic in Input-Queued Switches: New Delay Bounds via Probabilistic Techniques

Milan Vojnovic

n

th token that appears corresponds to permutation matrix

M_k

then we schedule matrix

M_k

in the

n

2002

n

th token that appears corresponds to permutation matrix

M_k

then we schedule matrix

M_k

in the

n

2003

n

th token that appears corresponds to permutation matrix

M_k

then we schedule matrix

M_k

in the

n

2003

n

th token that appears corresponds to permutation matrix

M_k

then we schedule matrix

M_k

in the

n

2003

Scheduling Reserved Traffic in Input-Queued Switches: New Delay Bounds via Probabilistic Techniques

Milan Vojnovic

n

th token that appears corresponds to permutation matrix

M_k

then we schedule matrix

M_k

in the

n

2002